Animated Solution for Physics - Properties of Solids and Liquids: Variation of radiant energy emitted by sun, filament of tungsten lamp and welding arc as a function of its wavelength is shown in figure. Which of the following option is the correct match?

The study of black-body radiation is one of the most fascinating chapters in physics, bridging the gap between classical thermodynamics and quantum mechanics. In this problem, we are presented with the spectral emission curves of three different real-world sources: the Sun, a tungsten filament, and a welding arc. Our goal is to match each source to its corresponding curve based on its temperature.

When we look at the graph of radiant energy ($E_{\lambda }$) versus wavelength ($\lambda$), we immediately notice three distinct curves labeled $T_1$, $T_2$, and $T_3$. The most crucial feature of these curves is their peak—the wavelength at which the source emits the maximum amount of energy, denoted as ${\lambda }_m$.

To make sense of this, we invoke Wien's Displacement Law, which states: where $b$ is Wien's displacement constant. This elegant equation tells us that the peak wavelength is inversely proportional to the absolute temperature of the black body. In simpler terms: the hotter the object, the shorter (and bluer) the wavelength at which it emits the most light.

By observing the graph, we can easily compare the peak wavelengths of the three curves: Because of the inverse relationship dictated by Wien's Law, the order of their temperatures must be the exact opposite: This means that the curve $T_3$ corresponds to the hottest source, while $T_1$ corresponds to the coolest.

Now, let's evaluate our three sources: 1. The Sun: A massive nuclear fusion reactor in the sky, the Sun has a surface temperature of approximately $6000\text{ K}$. 2. Welding Arc: An electric arc used for welding metals typically reaches temperatures around $4000\text{ K}$ to $5000\text{ K}$ (though some can be hotter, in the context of standard comparisons, it is generally cooler than the Sun). 3. Tungsten Filament: The glowing wire inside a traditional incandescent light bulb operates at about $3000\text{ K}$.

Comparing these temperatures, it is clear that the Sun is the absolute hottest of the three. Therefore, the Sun must correspond to the highest temperature curve, $T_3$.

Looking at our options, only option (c) correctly pairs the Sun with $T_3$. By identifying the Sun as the hottest source, we uniquely arrive at the correct option without needing to debate the exact temperatures of the other two sources!

This problem beautifully illustrates how we can deduce the physical properties of distant or untouchable objects simply by analyzing the light they emit. Whether it's a light bulb in your room or a star millions of light-years away, the universal laws of thermodynamics apply equally to all!